Both fully and empirical Bayes inference produce shrinkage estimators which combine observations with prior information to achieve better estimation. Fully Bayes inference assumes the hyperparameters of the prior distribution have hyperprior distributions, which need to be integrated out; while empirical Bayes sets the parameters in the highest level of the hierarchical model with their most likely value. Thus, as often we cannot obtain the closed form of the posterior distribution, fully Bayes inference requires a sampling strategy to approximate this distribution. Correspondingly, this comes at a high computational cost, especially for high-dimensional data [1]}. Conversely, in empirical Bayes the hyperparameters are obtained by maximizing marginal likelihood function with much lower computational cost. Hence, by combining empirical Bayes with the NB GLM we can produce an estimator for the probability parameter, \(\theta _i\) , of the NB distribution which should efficiently handle both over-dispersion and short data lengths.
